Matrices

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2+x-1=0$. Consider the set $T=\{1, \alpha, \beta\}$. For a $3 \times 3$ matrix $M=\left(a_{i j}\right)_{3 \times 3}$, define $R_i=a_{i 1}+a_{i 2}+a_{i…

Paragraph: Read the following passage and answer the questions. Question: The value of $\left[\begin{array}{lll}3 & 2 & 0\end{array}\right]\left[\begin{array}{l}3 \\ 2 \\ 0\end{array}\right]$ is

Paragraph: Read the following passage and answer the questions. Question: The value of the elements of $U^{-1}$ is

Paragraph: Read the following passage and answer the questions. Question: The value of $|U|$ is

Let $\mathbb{R}$ denote the set of all real numbers and let $i = \sqrt{-1}$. Consider the matrices $S = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ and $T = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. Let $a, b,…

For real numbers $\alpha, \beta, \gamma, \delta$ and $\mu$, consider the matrix $M = \begin{bmatrix} \alpha & \dfrac{1}{\sqrt{2}} & -\dfrac{1}{\sqrt{2}} \\ \dfrac{1}{\sqrt{3}} & \beta & \dfrac{1}{\sqrt{3}} \\ \gamma &…

Consider the matrix $M = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. Let $p, q, r, s, a, b, c$ and $d$ be integers such that $M^{26} = \begin{bmatrix} p & q \\ r & s \end{bmatrix}$ and $\sum_{k=1}^{26} M^k =…

Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?

Let $I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$ and $P=\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)$. Let $Q=\left(\begin{array}{ll}\mathrm{x} & \mathrm{y} \\ \mathrm{z} & 4\end{array}\right)$…

Consider the matrix, $P=\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right)$ Let the transpose of a matrix $X$ be denoted by $\mathrm{X}^T$. Then the number of $3 \times 3$ invertible matrices…

Let $M=(a_{ij}),i,j\in\{1,2,3\}$, be the $3\times3$ matrix such that $a_{ij}=1$ if $j+1$ is divisible by $i$, otherwise $a_{ij}=0$. Then which of the following statements is(are) true?

If $M=(\frac{5}{2} \frac{3}{2} \\ -\frac{3}{2} -\frac{1}{2})$, then which of the following matrices is equal to $M^{2022}$ ?